ergo
template_lapack_larre.h
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1 /* Ergo, version 3.8, a program for linear scaling electronic structure
2  * calculations.
3  * Copyright (C) 2019 Elias Rudberg, Emanuel H. Rubensson, Pawel Salek,
4  * and Anastasia Kruchinina.
5  *
6  * This program is free software: you can redistribute it and/or modify
7  * it under the terms of the GNU General Public License as published by
8  * the Free Software Foundation, either version 3 of the License, or
9  * (at your option) any later version.
10  *
11  * This program is distributed in the hope that it will be useful,
12  * but WITHOUT ANY WARRANTY; without even the implied warranty of
13  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14  * GNU General Public License for more details.
15  *
16  * You should have received a copy of the GNU General Public License
17  * along with this program. If not, see <http://www.gnu.org/licenses/>.
18  *
19  * Primary academic reference:
20  * Ergo: An open-source program for linear-scaling electronic structure
21  * calculations,
22  * Elias Rudberg, Emanuel H. Rubensson, Pawel Salek, and Anastasia
23  * Kruchinina,
24  * SoftwareX 7, 107 (2018),
25  * <http://dx.doi.org/10.1016/j.softx.2018.03.005>
26  *
27  * For further information about Ergo, see <http://www.ergoscf.org>.
28  */
29 
30  /* This file belongs to the template_lapack part of the Ergo source
31  * code. The source files in the template_lapack directory are modified
32  * versions of files originally distributed as CLAPACK, see the
33  * Copyright/license notice in the file template_lapack/COPYING.
34  */
35 
36 
37 #ifndef TEMPLATE_LAPACK_LARRE_HEADER
38 #define TEMPLATE_LAPACK_LARRE_HEADER
39 
40 
41 #include "template_lapack_larrk.h"
42 #include "template_lapack_lasq2.h"
43 
44 
45 template<class Treal>
46 int template_lapack_larre(const char *range, const integer *n, Treal *vl,
47  Treal *vu, integer *il, integer *iu, Treal *d__, Treal
48  *e, Treal *e2, Treal *rtol1, Treal *rtol2, Treal *
49  spltol, integer *nsplit, integer *isplit, integer *m, Treal *w,
50  Treal *werr, Treal *wgap, integer *iblock, integer *indexw,
51  Treal *gers, Treal *pivmin, Treal *work, integer *
52  iwork, integer *info)
53 {
54  /* System generated locals */
55  integer i__1, i__2;
56  Treal d__1, d__2, d__3;
57 
58 
59  /* Local variables */
60  integer i__, j;
61  Treal s1, s2;
62  integer mb = 0; // EMANUEL COMMENT: initialize to get rid of compiler warning
63  Treal gl;
64  integer in, mm;
65  Treal gu;
66  integer cnt;
67  Treal eps, tau, tmp, rtl;
68  integer cnt1, cnt2;
69  Treal tmp1, eabs;
70  integer iend, jblk;
71  Treal eold;
72  integer indl;
73  Treal dmax__, emax;
74  integer wend = 0; // EMANUEL COMMENT: initialize to get rid of compiler warning
75  integer idum, indu;
76  Treal rtol;
77  integer iseed[4];
78  Treal avgap, sigma;
79  integer iinfo;
80  logical norep;
81  integer ibegin;
82  logical forceb;
83  integer irange = 0; // EMANUEL COMMENT: initialize to get rid of compiler warning
84  Treal sgndef;
85  integer wbegin;
86  Treal safmin, spdiam;
87  logical usedqd;
88  Treal clwdth, isleft;
89  Treal isrght, bsrtol, dpivot;
90 
91 
92 /* -- LAPACK auxiliary routine (version 3.2) -- */
93 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
94 /* November 2006 */
95 
96 /* .. Scalar Arguments .. */
97 /* .. */
98 /* .. Array Arguments .. */
99 /* .. */
100 
101 /* Purpose */
102 /* ======= */
103 
104 /* To find the desired eigenvalues of a given real symmetric */
105 /* tridiagonal matrix T, DLARRE sets any "small" off-diagonal */
106 /* elements to zero, and for each unreduced block T_i, it finds */
107 /* (a) a suitable shift at one end of the block's spectrum, */
108 /* (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and */
109 /* (c) eigenvalues of each L_i D_i L_i^T. */
110 /* The representations and eigenvalues found are then used by */
111 /* DSTEMR to compute the eigenvectors of T. */
112 /* The accuracy varies depending on whether bisection is used to */
113 /* find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to */
114 /* conpute all and then discard any unwanted one. */
115 /* As an added benefit, DLARRE also outputs the n */
116 /* Gerschgorin intervals for the matrices L_i D_i L_i^T. */
117 
118 /* Arguments */
119 /* ========= */
120 
121 /* RANGE (input) CHARACTER */
122 /* = 'A': ("All") all eigenvalues will be found. */
123 /* = 'V': ("Value") all eigenvalues in the half-open interval */
124 /* (VL, VU] will be found. */
125 /* = 'I': ("Index") the IL-th through IU-th eigenvalues (of the */
126 /* entire matrix) will be found. */
127 
128 /* N (input) INTEGER */
129 /* The order of the matrix. N > 0. */
130 
131 /* VL (input/output) DOUBLE PRECISION */
132 /* VU (input/output) DOUBLE PRECISION */
133 /* If RANGE='V', the lower and upper bounds for the eigenvalues. */
134 /* Eigenvalues less than or equal to VL, or greater than VU, */
135 /* will not be returned. VL < VU. */
136 /* If RANGE='I' or ='A', DLARRE computes bounds on the desired */
137 /* part of the spectrum. */
138 
139 /* IL (input) INTEGER */
140 /* IU (input) INTEGER */
141 /* If RANGE='I', the indices (in ascending order) of the */
142 /* smallest and largest eigenvalues to be returned. */
143 /* 1 <= IL <= IU <= N. */
144 
145 /* D (input/output) DOUBLE PRECISION array, dimension (N) */
146 /* On entry, the N diagonal elements of the tridiagonal */
147 /* matrix T. */
148 /* On exit, the N diagonal elements of the diagonal */
149 /* matrices D_i. */
150 
151 /* E (input/output) DOUBLE PRECISION array, dimension (N) */
152 /* On entry, the first (N-1) entries contain the subdiagonal */
153 /* elements of the tridiagonal matrix T; E(N) need not be set. */
154 /* On exit, E contains the subdiagonal elements of the unit */
155 /* bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), */
156 /* 1 <= I <= NSPLIT, contain the base points sigma_i on output. */
157 
158 /* E2 (input/output) DOUBLE PRECISION array, dimension (N) */
159 /* On entry, the first (N-1) entries contain the SQUARES of the */
160 /* subdiagonal elements of the tridiagonal matrix T; */
161 /* E2(N) need not be set. */
162 /* On exit, the entries E2( ISPLIT( I ) ), */
163 /* 1 <= I <= NSPLIT, have been set to zero */
164 
165 /* RTOL1 (input) DOUBLE PRECISION */
166 /* RTOL2 (input) DOUBLE PRECISION */
167 /* Parameters for bisection. */
168 /* An interval [LEFT,RIGHT] has converged if */
169 /* RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) */
170 
171 /* SPLTOL (input) DOUBLE PRECISION */
172 /* The threshold for splitting. */
173 
174 /* NSPLIT (output) INTEGER */
175 /* The number of blocks T splits into. 1 <= NSPLIT <= N. */
176 
177 /* ISPLIT (output) INTEGER array, dimension (N) */
178 /* The splitting points, at which T breaks up into blocks. */
179 /* The first block consists of rows/columns 1 to ISPLIT(1), */
180 /* the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), */
181 /* etc., and the NSPLIT-th consists of rows/columns */
182 /* ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. */
183 
184 /* M (output) INTEGER */
185 /* The total number of eigenvalues (of all L_i D_i L_i^T) */
186 /* found. */
187 
188 /* W (output) DOUBLE PRECISION array, dimension (N) */
189 /* The first M elements contain the eigenvalues. The */
190 /* eigenvalues of each of the blocks, L_i D_i L_i^T, are */
191 /* sorted in ascending order ( DLARRE may use the */
192 /* remaining N-M elements as workspace). */
193 
194 /* WERR (output) DOUBLE PRECISION array, dimension (N) */
195 /* The error bound on the corresponding eigenvalue in W. */
196 
197 /* WGAP (output) DOUBLE PRECISION array, dimension (N) */
198 /* The separation from the right neighbor eigenvalue in W. */
199 /* The gap is only with respect to the eigenvalues of the same block */
200 /* as each block has its own representation tree. */
201 /* Exception: at the right end of a block we store the left gap */
202 
203 /* IBLOCK (output) INTEGER array, dimension (N) */
204 /* The indices of the blocks (submatrices) associated with the */
205 /* corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue */
206 /* W(i) belongs to the first block from the top, =2 if W(i) */
207 /* belongs to the second block, etc. */
208 
209 /* INDEXW (output) INTEGER array, dimension (N) */
210 /* The indices of the eigenvalues within each block (submatrix); */
211 /* for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the */
212 /* i-th eigenvalue W(i) is the 10-th eigenvalue in block 2 */
213 
214 /* GERS (output) DOUBLE PRECISION array, dimension (2*N) */
215 /* The N Gerschgorin intervals (the i-th Gerschgorin interval */
216 /* is (GERS(2*i-1), GERS(2*i)). */
217 
218 /* PIVMIN (output) DOUBLE PRECISION */
219 /* The minimum pivot in the Sturm sequence for T. */
220 
221 /* WORK (workspace) DOUBLE PRECISION array, dimension (6*N) */
222 /* Workspace. */
223 
224 /* IWORK (workspace) INTEGER array, dimension (5*N) */
225 /* Workspace. */
226 
227 /* INFO (output) INTEGER */
228 /* = 0: successful exit */
229 /* > 0: A problem occured in DLARRE. */
230 /* < 0: One of the called subroutines signaled an internal problem. */
231 /* Needs inspection of the corresponding parameter IINFO */
232 /* for further information. */
233 
234 /* =-1: Problem in DLARRD. */
235 /* = 2: No base representation could be found in MAXTRY iterations. */
236 /* Increasing MAXTRY and recompilation might be a remedy. */
237 /* =-3: Problem in DLARRB when computing the refined root */
238 /* representation for DLASQ2. */
239 /* =-4: Problem in DLARRB when preforming bisection on the */
240 /* desired part of the spectrum. */
241 /* =-5: Problem in DLASQ2. */
242 /* =-6: Problem in DLASQ2. */
243 
244 /* Further Details */
245 /* The base representations are required to suffer very little */
246 /* element growth and consequently define all their eigenvalues to */
247 /* high relative accuracy. */
248 /* =============== */
249 
250 /* Based on contributions by */
251 /* Beresford Parlett, University of California, Berkeley, USA */
252 /* Jim Demmel, University of California, Berkeley, USA */
253 /* Inderjit Dhillon, University of Texas, Austin, USA */
254 /* Osni Marques, LBNL/NERSC, USA */
255 /* Christof Voemel, University of California, Berkeley, USA */
256 
257 /* ===================================================================== */
258 
259 /* .. Parameters .. */
260 /* .. */
261 /* .. Local Scalars .. */
262 /* .. */
263 /* .. Local Arrays .. */
264 /* .. */
265 /* .. External Functions .. */
266 /* .. */
267 /* .. External Subroutines .. */
268 /* .. */
269 /* .. Intrinsic Functions .. */
270 /* .. */
271 /* .. Executable Statements .. */
272 
273  /* Parameter adjustments */
274 
275 
276  /* Table of constant values */
277 
278  integer c__1 = 1;
279  integer c__2 = 2;
280 
281 
282  --iwork;
283  --work;
284  --gers;
285  --indexw;
286  --iblock;
287  --wgap;
288  --werr;
289  --w;
290  --isplit;
291  --e2;
292  --e;
293  --d__;
294 
295  /* Initialization added by Elias to get rid of compiler warnings. */
296  mm = 0;
297  /* Function Body */
298  *info = 0;
299 
300 /* Decode RANGE */
301 
302  if (template_blas_lsame(range, "A")) {
303  irange = 1;
304  } else if (template_blas_lsame(range, "V")) {
305  irange = 3;
306  } else if (template_blas_lsame(range, "I")) {
307  irange = 2;
308  }
309  *m = 0;
310 /* Get machine constants */
311  safmin = template_lapack_lamch("S",(Treal)0);
312  eps = template_lapack_lamch("P",(Treal)0);
313 /* Set parameters */
314  rtl = template_blas_sqrt(eps);
315  bsrtol = template_blas_sqrt(eps);
316 /* Treat case of 1x1 matrix for quick return */
317  if (*n == 1) {
318  if (irange == 1 || ( irange == 3 && d__[1] > *vl && d__[1] <= *vu ) ||
319  ( irange == 2 && *il == 1 && *iu == 1 ) ) {
320  *m = 1;
321  w[1] = d__[1];
322 /* The computation error of the eigenvalue is zero */
323  werr[1] = 0.;
324  wgap[1] = 0.;
325  iblock[1] = 1;
326  indexw[1] = 1;
327  gers[1] = d__[1];
328  gers[2] = d__[1];
329  }
330 /* store the shift for the initial RRR, which is zero in this case */
331  e[1] = 0.;
332  return 0;
333  }
334 /* General case: tridiagonal matrix of order > 1 */
335 
336 /* Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter. */
337 /* Compute maximum off-diagonal entry and pivmin. */
338  gl = d__[1];
339  gu = d__[1];
340  eold = 0.;
341  emax = 0.;
342  e[*n] = 0.;
343  i__1 = *n;
344  for (i__ = 1; i__ <= i__1; ++i__) {
345  werr[i__] = 0.;
346  wgap[i__] = 0.;
347  eabs = (d__1 = e[i__], absMACRO(d__1));
348  if (eabs >= emax) {
349  emax = eabs;
350  }
351  tmp1 = eabs + eold;
352  gers[(i__ << 1) - 1] = d__[i__] - tmp1;
353 /* Computing MIN */
354  d__1 = gl, d__2 = gers[(i__ << 1) - 1];
355  gl = minMACRO(d__1,d__2);
356  gers[i__ * 2] = d__[i__] + tmp1;
357 /* Computing MAX */
358  d__1 = gu, d__2 = gers[i__ * 2];
359  gu = maxMACRO(d__1,d__2);
360  eold = eabs;
361 /* L5: */
362  }
363 /* The minimum pivot allowed in the Sturm sequence for T */
364 /* Computing MAX */
365 /* Computing 2nd power */
366  d__3 = emax;
367  d__1 = 1., d__2 = d__3 * d__3;
368  *pivmin = safmin * maxMACRO(d__1,d__2);
369 /* Compute spectral diameter. The Gerschgorin bounds give an */
370 /* estimate that is wrong by at most a factor of SQRT(2) */
371  spdiam = gu - gl;
372 /* Compute splitting points */
373  template_lapack_larra(n, &d__[1], &e[1], &e2[1], spltol, &spdiam, nsplit, &isplit[1], &
374  iinfo);
375 /* Can force use of bisection instead of faster DQDS. */
376 /* Option left in the code for future multisection work. */
377  forceb = FALSE_;
378 /* Initialize USEDQD, DQDS should be used for ALLRNG unless someone */
379 /* explicitly wants bisection. */
380  usedqd = irange == 1 && ! forceb;
381  if (irange == 1 && ! forceb) {
382 /* Set interval [VL,VU] that contains all eigenvalues */
383  *vl = gl;
384  *vu = gu;
385  } else {
386 /* We call DLARRD to find crude approximations to the eigenvalues */
387 /* in the desired range. In case IRANGE = INDRNG, we also obtain the */
388 /* interval (VL,VU] that contains all the wanted eigenvalues. */
389 /* An interval [LEFT,RIGHT] has converged if */
390 /* RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT)) */
391 /* DLARRD needs a WORK of size 4*N, IWORK of size 3*N */
392  template_lapack_larrd(range, "B", n, vl, vu, il, iu, &gers[1], &bsrtol, &d__[1], &e[
393  1], &e2[1], pivmin, nsplit, &isplit[1], &mm, &w[1], &werr[1],
394  vl, vu, &iblock[1], &indexw[1], &work[1], &iwork[1], &iinfo);
395  if (iinfo != 0) {
396  *info = -1;
397  return 0;
398  }
399 /* Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0 */
400  i__1 = *n;
401  for (i__ = mm + 1; i__ <= i__1; ++i__) {
402  w[i__] = 0.;
403  werr[i__] = 0.;
404  iblock[i__] = 0;
405  indexw[i__] = 0;
406 /* L14: */
407  }
408  }
409 /* ** */
410 /* Loop over unreduced blocks */
411  ibegin = 1;
412  wbegin = 1;
413  i__1 = *nsplit;
414  for (jblk = 1; jblk <= i__1; ++jblk) {
415  iend = isplit[jblk];
416  in = iend - ibegin + 1;
417 /* 1 X 1 block */
418  if (in == 1) {
419  if (irange == 1 || ( irange == 3 && d__[ibegin] > *vl && d__[ibegin]
420  <= *vu ) || ( irange == 2 && iblock[wbegin] == jblk ) ) {
421  ++(*m);
422  w[*m] = d__[ibegin];
423  werr[*m] = 0.;
424 /* The gap for a single block doesn't matter for the later */
425 /* algorithm and is assigned an arbitrary large value */
426  wgap[*m] = 0.;
427  iblock[*m] = jblk;
428  indexw[*m] = 1;
429  ++wbegin;
430  }
431 /* E( IEND ) holds the shift for the initial RRR */
432  e[iend] = 0.;
433  ibegin = iend + 1;
434  goto L170;
435  }
436 
437 /* Blocks of size larger than 1x1 */
438 
439 /* E( IEND ) will hold the shift for the initial RRR, for now set it =0 */
440  e[iend] = 0.;
441 
442 /* Find local outer bounds GL,GU for the block */
443  gl = d__[ibegin];
444  gu = d__[ibegin];
445  i__2 = iend;
446  for (i__ = ibegin; i__ <= i__2; ++i__) {
447 /* Computing MIN */
448  d__1 = gers[(i__ << 1) - 1];
449  gl = minMACRO(d__1,gl);
450 /* Computing MAX */
451  d__1 = gers[i__ * 2];
452  gu = maxMACRO(d__1,gu);
453 /* L15: */
454  }
455  spdiam = gu - gl;
456  if (! (irange == 1 && ! forceb)) {
457 /* Count the number of eigenvalues in the current block. */
458  mb = 0;
459  i__2 = mm;
460  for (i__ = wbegin; i__ <= i__2; ++i__) {
461  if (iblock[i__] == jblk) {
462  ++mb;
463  } else {
464  goto L21;
465  }
466 /* L20: */
467  }
468 L21:
469  if (mb == 0) {
470 /* No eigenvalue in the current block lies in the desired range */
471 /* E( IEND ) holds the shift for the initial RRR */
472  e[iend] = 0.;
473  ibegin = iend + 1;
474  goto L170;
475  } else {
476 /* Decide whether dqds or bisection is more efficient */
477  usedqd = (Treal) mb > in * .5 && ! forceb;
478  wend = wbegin + mb - 1;
479 /* Calculate gaps for the current block */
480 /* In later stages, when representations for individual */
481 /* eigenvalues are different, we use SIGMA = E( IEND ). */
482  sigma = 0.;
483  i__2 = wend - 1;
484  for (i__ = wbegin; i__ <= i__2; ++i__) {
485 /* Computing MAX */
486  d__1 = 0., d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
487  werr[i__]);
488  wgap[i__] = maxMACRO(d__1,d__2);
489 /* L30: */
490  }
491 /* Computing MAX */
492  d__1 = 0., d__2 = *vu - sigma - (w[wend] + werr[wend]);
493  wgap[wend] = maxMACRO(d__1,d__2);
494 /* Find local index of the first and last desired evalue. */
495  indl = indexw[wbegin];
496  indu = indexw[wend];
497  }
498  }
499  if ( ( irange == 1 && ! forceb ) || usedqd) {
500 /* Case of DQDS */
501 /* Find approximations to the extremal eigenvalues of the block */
502  template_lapack_larrk(&in, &c__1, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
503  rtl, &tmp, &tmp1, &iinfo);
504  if (iinfo != 0) {
505  *info = -1;
506  return 0;
507  }
508 /* Computing MAX */
509  d__2 = gl, d__3 = tmp - tmp1 - eps * 100. * (d__1 = tmp - tmp1,
510  absMACRO(d__1));
511  isleft = maxMACRO(d__2,d__3);
512  template_lapack_larrk(&in, &in, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
513  rtl, &tmp, &tmp1, &iinfo);
514  if (iinfo != 0) {
515  *info = -1;
516  return 0;
517  }
518 /* Computing MIN */
519  d__2 = gu, d__3 = tmp + tmp1 + eps * 100. * (d__1 = tmp + tmp1,
520  absMACRO(d__1));
521  isrght = minMACRO(d__2,d__3);
522 /* Improve the estimate of the spectral diameter */
523  spdiam = isrght - isleft;
524  } else {
525 /* Case of bisection */
526 /* Find approximations to the wanted extremal eigenvalues */
527 /* Computing MAX */
528  d__2 = gl, d__3 = w[wbegin] - werr[wbegin] - eps * 100. * (d__1 =
529  w[wbegin] - werr[wbegin], absMACRO(d__1));
530  isleft = maxMACRO(d__2,d__3);
531 /* Computing MIN */
532  d__2 = gu, d__3 = w[wend] + werr[wend] + eps * 100. * (d__1 = w[
533  wend] + werr[wend], absMACRO(d__1));
534  isrght = minMACRO(d__2,d__3);
535  }
536 /* Decide whether the base representation for the current block */
537 /* L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I */
538 /* should be on the left or the right end of the current block. */
539 /* The strategy is to shift to the end which is "more populated" */
540 /* Furthermore, decide whether to use DQDS for the computation of */
541 /* the eigenvalue approximations at the end of DLARRE or bisection. */
542 /* dqds is chosen if all eigenvalues are desired or the number of */
543 /* eigenvalues to be computed is large compared to the blocksize. */
544  if (irange == 1 && ! forceb) {
545 /* If all the eigenvalues have to be computed, we use dqd */
546  usedqd = TRUE_;
547 /* INDL is the local index of the first eigenvalue to compute */
548  indl = 1;
549  indu = in;
550 /* MB = number of eigenvalues to compute */
551  mb = in;
552  wend = wbegin + mb - 1;
553 /* Define 1/4 and 3/4 points of the spectrum */
554  s1 = isleft + spdiam * .25;
555  s2 = isrght - spdiam * .25;
556  } else {
557 /* DLARRD has computed IBLOCK and INDEXW for each eigenvalue */
558 /* approximation. */
559 /* choose sigma */
560  if (usedqd) {
561  s1 = isleft + spdiam * .25;
562  s2 = isrght - spdiam * .25;
563  } else {
564  tmp = minMACRO(isrght,*vu) - maxMACRO(isleft,*vl);
565  s1 = maxMACRO(isleft,*vl) + tmp * .25;
566  s2 = minMACRO(isrght,*vu) - tmp * .25;
567  }
568  }
569 /* Compute the negcount at the 1/4 and 3/4 points */
570  if (mb > 1) {
571  template_lapack_larrc("T", &in, &s1, &s2, &d__[ibegin], &e[ibegin], pivmin, &
572  cnt, &cnt1, &cnt2, &iinfo);
573  }
574  if (mb == 1) {
575  sigma = gl;
576  sgndef = 1.;
577  } else if (cnt1 - indl >= indu - cnt2) {
578  if (irange == 1 && ! forceb) {
579  sigma = maxMACRO(isleft,gl);
580  } else if (usedqd) {
581 /* use Gerschgorin bound as shift to get pos def matrix */
582 /* for dqds */
583  sigma = isleft;
584  } else {
585 /* use approximation of the first desired eigenvalue of the */
586 /* block as shift */
587  sigma = maxMACRO(isleft,*vl);
588  }
589  sgndef = 1.;
590  } else {
591  if (irange == 1 && ! forceb) {
592  sigma = minMACRO(isrght,gu);
593  } else if (usedqd) {
594 /* use Gerschgorin bound as shift to get neg def matrix */
595 /* for dqds */
596  sigma = isrght;
597  } else {
598 /* use approximation of the first desired eigenvalue of the */
599 /* block as shift */
600  sigma = minMACRO(isrght,*vu);
601  }
602  sgndef = -1.;
603  }
604 /* An initial SIGMA has been chosen that will be used for computing */
605 /* T - SIGMA I = L D L^T */
606 /* Define the increment TAU of the shift in case the initial shift */
607 /* needs to be refined to obtain a factorization with not too much */
608 /* element growth. */
609  if (usedqd) {
610 /* The initial SIGMA was to the outer end of the spectrum */
611 /* the matrix is definite and we need not retreat. */
612  tau = spdiam * eps * *n + *pivmin * 2.;
613  } else {
614  if (mb > 1) {
615  clwdth = w[wend] + werr[wend] - w[wbegin] - werr[wbegin];
616  avgap = (d__1 = clwdth / (Treal) (wend - wbegin), absMACRO(
617  d__1));
618  if (sgndef == 1.) {
619 /* Computing MAX */
620  d__1 = wgap[wbegin];
621  tau = maxMACRO(d__1,avgap) * .5;
622 /* Computing MAX */
623  d__1 = tau, d__2 = werr[wbegin];
624  tau = maxMACRO(d__1,d__2);
625  } else {
626 /* Computing MAX */
627  d__1 = wgap[wend - 1];
628  tau = maxMACRO(d__1,avgap) * .5;
629 /* Computing MAX */
630  d__1 = tau, d__2 = werr[wend];
631  tau = maxMACRO(d__1,d__2);
632  }
633  } else {
634  tau = werr[wbegin];
635  }
636  }
637 
638  for (idum = 1; idum <= 6; ++idum) {
639 /* Compute L D L^T factorization of tridiagonal matrix T - sigma I. */
640 /* Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of */
641 /* pivots in WORK(2*IN+1:3*IN) */
642  dpivot = d__[ibegin] - sigma;
643  work[1] = dpivot;
644  dmax__ = absMACRO(work[1]);
645  j = ibegin;
646  i__2 = in - 1;
647  for (i__ = 1; i__ <= i__2; ++i__) {
648  work[(in << 1) + i__] = 1. / work[i__];
649  tmp = e[j] * work[(in << 1) + i__];
650  work[in + i__] = tmp;
651  dpivot = d__[j + 1] - sigma - tmp * e[j];
652  work[i__ + 1] = dpivot;
653 /* Computing MAX */
654  d__1 = dmax__, d__2 = absMACRO(dpivot);
655  dmax__ = maxMACRO(d__1,d__2);
656  ++j;
657 /* L70: */
658  }
659 /* check for element growth */
660  if (dmax__ > spdiam * 64.) {
661  norep = TRUE_;
662  } else {
663  norep = FALSE_;
664  }
665  if (usedqd && ! norep) {
666 /* Ensure the definiteness of the representation */
667 /* All entries of D (of L D L^T) must have the same sign */
668  i__2 = in;
669  for (i__ = 1; i__ <= i__2; ++i__) {
670  tmp = sgndef * work[i__];
671  if (tmp < 0.) {
672  norep = TRUE_;
673  }
674 /* L71: */
675  }
676  }
677  if (norep) {
678 /* Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin */
679 /* shift which makes the matrix definite. So we should end up */
680 /* here really only in the case of IRANGE = VALRNG or INDRNG. */
681  if (idum == 5) {
682  if (sgndef == 1.) {
683 /* The fudged Gerschgorin shift should succeed */
684  sigma = gl - spdiam * 2. * eps * *n - *pivmin * 4.;
685  } else {
686  sigma = gu + spdiam * 2. * eps * *n + *pivmin * 4.;
687  }
688  } else {
689  sigma -= sgndef * tau;
690  tau *= 2.;
691  }
692  } else {
693 /* an initial RRR is found */
694  goto L83;
695  }
696 /* L80: */
697  }
698 /* if the program reaches this point, no base representation could be */
699 /* found in MAXTRY iterations. */
700  *info = 2;
701  return 0;
702 L83:
703 /* At this point, we have found an initial base representation */
704 /* T - SIGMA I = L D L^T with not too much element growth. */
705 /* Store the shift. */
706  e[iend] = sigma;
707 /* Store D and L. */
708  template_blas_copy(&in, &work[1], &c__1, &d__[ibegin], &c__1);
709  i__2 = in - 1;
710  template_blas_copy(&i__2, &work[in + 1], &c__1, &e[ibegin], &c__1);
711  if (mb > 1) {
712 
713 /* Perturb each entry of the base representation by a small */
714 /* (but random) relative amount to overcome difficulties with */
715 /* glued matrices. */
716 
717  for (i__ = 1; i__ <= 4; ++i__) {
718  iseed[i__ - 1] = 1;
719 /* L122: */
720  }
721  i__2 = (in << 1) - 1;
722  template_lapack_larnv(&c__2, iseed, &i__2, &work[1]);
723  i__2 = in - 1;
724  for (i__ = 1; i__ <= i__2; ++i__) {
725  d__[ibegin + i__ - 1] *= eps * 8. * work[i__] + 1.;
726  e[ibegin + i__ - 1] *= eps * 8. * work[in + i__] + 1.;
727 /* L125: */
728  }
729  d__[iend] *= eps * 4. * work[in] + 1.;
730 
731  }
732 
733 /* Don't update the Gerschgorin intervals because keeping track */
734 /* of the updates would be too much work in DLARRV. */
735 /* We update W instead and use it to locate the proper Gerschgorin */
736 /* intervals. */
737 /* Compute the required eigenvalues of L D L' by bisection or dqds */
738  if (! usedqd) {
739 /* If DLARRD has been used, shift the eigenvalue approximations */
740 /* according to their representation. This is necessary for */
741 /* a uniform DLARRV since dqds computes eigenvalues of the */
742 /* shifted representation. In DLARRV, W will always hold the */
743 /* UNshifted eigenvalue approximation. */
744  i__2 = wend;
745  for (j = wbegin; j <= i__2; ++j) {
746  w[j] -= sigma;
747  werr[j] += (d__1 = w[j], absMACRO(d__1)) * eps;
748 /* L134: */
749  }
750 /* call DLARRB to reduce eigenvalue error of the approximations */
751 /* from DLARRD */
752  i__2 = iend - 1;
753  for (i__ = ibegin; i__ <= i__2; ++i__) {
754 /* Computing 2nd power */
755  d__1 = e[i__];
756  work[i__] = d__[i__] * (d__1 * d__1);
757 /* L135: */
758  }
759 /* use bisection to find EV from INDL to INDU */
760  i__2 = indl - 1;
761  template_lapack_larrb(&in, &d__[ibegin], &work[ibegin], &indl, &indu, rtol1,
762  rtol2, &i__2, &w[wbegin], &wgap[wbegin], &werr[wbegin], &
763  work[(*n << 1) + 1], &iwork[1], pivmin, &spdiam, &in, &
764  iinfo);
765  if (iinfo != 0) {
766  *info = -4;
767  return 0;
768  }
769 /* DLARRB computes all gaps correctly except for the last one */
770 /* Record distance to VU/GU */
771 /* Computing MAX */
772  d__1 = 0., d__2 = *vu - sigma - (w[wend] + werr[wend]);
773  wgap[wend] = maxMACRO(d__1,d__2);
774  i__2 = indu;
775  for (i__ = indl; i__ <= i__2; ++i__) {
776  ++(*m);
777  iblock[*m] = jblk;
778  indexw[*m] = i__;
779 /* L138: */
780  }
781  } else {
782 /* Call dqds to get all eigs (and then possibly delete unwanted */
783 /* eigenvalues). */
784 /* Note that dqds finds the eigenvalues of the L D L^T representation */
785 /* of T to high relative accuracy. High relative accuracy */
786 /* might be lost when the shift of the RRR is subtracted to obtain */
787 /* the eigenvalues of T. However, T is not guaranteed to define its */
788 /* eigenvalues to high relative accuracy anyway. */
789 /* Set RTOL to the order of the tolerance used in DLASQ2 */
790 /* This is an ESTIMATED error, the worst case bound is 4*N*EPS */
791 /* which is usually too large and requires unnecessary work to be */
792 /* done by bisection when computing the eigenvectors */
793  rtol = template_blas_log((Treal) in) * 4. * eps;
794  j = ibegin;
795  i__2 = in - 1;
796  for (i__ = 1; i__ <= i__2; ++i__) {
797  work[(i__ << 1) - 1] = (d__1 = d__[j], absMACRO(d__1));
798  work[i__ * 2] = e[j] * e[j] * work[(i__ << 1) - 1];
799  ++j;
800 /* L140: */
801  }
802  work[(in << 1) - 1] = (d__1 = d__[iend], absMACRO(d__1));
803  work[in * 2] = 0.;
804  template_lapack_lasq2(&in, &work[1], &iinfo);
805  if (iinfo != 0) {
806 /* If IINFO = -5 then an index is part of a tight cluster */
807 /* and should be changed. The index is in IWORK(1) and the */
808 /* gap is in WORK(N+1) */
809  *info = -5;
810  return 0;
811  } else {
812 /* Test that all eigenvalues are positive as expected */
813  i__2 = in;
814  for (i__ = 1; i__ <= i__2; ++i__) {
815  if (work[i__] < 0.) {
816  *info = -6;
817  return 0;
818  }
819 /* L149: */
820  }
821  }
822  if (sgndef > 0.) {
823  i__2 = indu;
824  for (i__ = indl; i__ <= i__2; ++i__) {
825  ++(*m);
826  w[*m] = work[in - i__ + 1];
827  iblock[*m] = jblk;
828  indexw[*m] = i__;
829 /* L150: */
830  }
831  } else {
832  i__2 = indu;
833  for (i__ = indl; i__ <= i__2; ++i__) {
834  ++(*m);
835  w[*m] = -work[i__];
836  iblock[*m] = jblk;
837  indexw[*m] = i__;
838 /* L160: */
839  }
840  }
841  i__2 = *m;
842  for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
843 /* the value of RTOL below should be the tolerance in DLASQ2 */
844  werr[i__] = rtol * (d__1 = w[i__], absMACRO(d__1));
845 /* L165: */
846  }
847  i__2 = *m - 1;
848  for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
849 /* compute the right gap between the intervals */
850 /* Computing MAX */
851  d__1 = 0., d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] + werr[
852  i__]);
853  wgap[i__] = maxMACRO(d__1,d__2);
854 /* L166: */
855  }
856 /* Computing MAX */
857  d__1 = 0., d__2 = *vu - sigma - (w[*m] + werr[*m]);
858  wgap[*m] = maxMACRO(d__1,d__2);
859  }
860 /* proceed with next block */
861  ibegin = iend + 1;
862  wbegin = wend + 1;
863 L170:
864  ;
865  }
866 
867  return 0;
868 
869 /* end of DLARRE */
870 
871 } /* dlarre_ */
872 
873 #endif
template_blas_sqrt
Treal template_blas_sqrt(Treal x)
gu
static const real gu
Definition: fun-pz81.c:68
template_lapack_lasq2.h
template_lapack_larrd
int template_lapack_larrd(const char *range, const char *order, const integer *n, Treal *vl, Treal *vu, integer *il, integer *iu, Treal *gers, Treal *reltol, Treal *d__, Treal *e, Treal *e2, Treal *pivmin, integer *nsplit, integer *isplit, integer *m, Treal *w, Treal *werr, Treal *wl, Treal *wu, integer *iblock, integer *indexw, Treal *work, integer *iwork, integer *info)
Definition: template_lapack_larrd.h:41
template_lapack_lamch
Treal template_lapack_lamch(const char *cmach, Treal dummyReal)
Definition: template_lapack_lamch.h:202
template_lapack_larrk.h
minMACRO
#define minMACRO(a, b)
Definition: template_blas_common.h:46
template_lapack_larnv
int template_lapack_larnv(const integer *idist, integer *iseed, const integer *n, Treal *x)
Definition: template_lapack_larnv.h:42
absMACRO
#define absMACRO(x)
Definition: template_blas_common.h:47
template_lapack_larrc
int template_lapack_larrc(const char *jobt, const integer *n, const Treal *vl, const Treal *vu, Treal *d__, Treal *e, Treal *pivmin, integer *eigcnt, integer *lcnt, integer *rcnt, integer *info)
Definition: template_lapack_larrc.h:41
logical
bool logical
Definition: template_blas_common.h:41
template_lapack_larrk
int template_lapack_larrk(integer *n, integer *iw, Treal *gl, Treal *gu, Treal *d__, Treal *e2, Treal *pivmin, Treal *reltol, Treal *w, Treal *werr, integer *info)
Definition: template_lapack_larrk.h:41
template_lapack_larra
int template_lapack_larra(const integer *n, Treal *d__, Treal *e, Treal *e2, Treal *spltol, Treal *tnrm, integer *nsplit, integer *isplit, integer *info)
Definition: template_lapack_larra.h:41
template_lapack_larrb
int template_lapack_larrb(integer *n, Treal *d__, Treal *lld, integer *ifirst, integer *ilast, Treal *rtol1, Treal *rtol2, integer *offset, Treal *w, Treal *wgap, Treal *werr, Treal *work, integer *iwork, Treal *pivmin, Treal *spdiam, integer *twist, integer *info)
Definition: template_lapack_larrb.h:45
template_blas_log
Treal template_blas_log(Treal x)
template_blas_copy
int template_blas_copy(const integer *n, const Treal *dx, const integer *incx, Treal *dy, const integer *incy)
Definition: template_blas_copy.h:42
template_blas_lsame
logical template_blas_lsame(const char *ca, const char *cb)
Definition: template_blas_common.cc:46
TRUE_
#define TRUE_
Definition: template_lapack_common.h:42
integer
int integer
Definition: template_blas_common.h:40
template_lapack_lasq2
int template_lapack_lasq2(integer *n, Treal *z__, integer *info)
Definition: template_lapack_lasq2.h:45
FALSE_
#define FALSE_
Definition: template_lapack_common.h:43
maxMACRO
#define maxMACRO(a, b)
Definition: template_blas_common.h:45
template_lapack_larre
int template_lapack_larre(const char *range, const integer *n, Treal *vl, Treal *vu, integer *il, integer *iu, Treal *d__, Treal *e, Treal *e2, Treal *rtol1, Treal *rtol2, Treal *spltol, integer *nsplit, integer *isplit, integer *m, Treal *w, Treal *werr, Treal *wgap, integer *iblock, integer *indexw, Treal *gers, Treal *pivmin, Treal *work, integer *iwork, integer *info)
Definition: template_lapack_larre.h:46